Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- 2*(cos(z)+cos(3*z))/2*sin(2*z)+sin(4*z) если z=-1/3
- cos(2*a)*cos(3*a)-sin(2*a)*sin(3*a) если a=3
- a*sqrt(a)+b*sqrt(b)+a*sqrt(b)+b*sqrt(a) если a=-2
- cos(pi+3*t) если t=1
- Разложить многочлен на множители:
- x^2-5
- 2-y^2
- c^2-81
- C^2-9
- Общий знаменатель:
- a+3/a-1-a/1-a
- a+3/a*2+3*a+9-1/a-3+a*3+3*a-9/a*3-27
- (25-3*x)/(x-5)^2-(7*x-x^2)/(5-x)^2
- (((x-y)/(sqrt(x)-sqrt(y)))-((x-sqrt(x*y))/(sqrt(x)-sqrt(y))))^(-2)
- Умножение столбиком:
- 14*25
- 12*60
- 20*16
- 12*24
- Деление столбиком:
- 18018/6
- 45/5
- 29/5
- 34/8
- Раскрыть скобки в:
- 2*(z+x)*3
- (x-2)^4
- (a+2)*(a+2)^2
- (p+q)^2*(p-q)^2
- Разложение числа на простые множители:
- 2002
- 7140
- 4982
- 4356
- Производная:
- -1/3
- Интеграл d{x}:
-
-1/3
-
Идентичные выражения
- два *(cos(z)+cos(три *z))/ два *sin(два *z)+sin(четыре *z) если z=- один / три
- 2 умножить на ( косинус от (z) плюс косинус от (3 умножить на z)) делить на 2 умножить на синус от (2 умножить на z) плюс синус от (4 умножить на z) если z равно минус 1 делить на 3
- два умножить на ( косинус от (z) плюс косинус от (три умножить на z)) делить на два умножить на синус от (два умножить на z) плюс синус от (четыре умножить на z) если z равно минус один делить на три
- 2(cos(z)+cos(3z))/2sin(2z)+sin(4z) если z=-1/3
- 2cosz+cos3z/2sin2z+sin4z если z=-1/3
- 2*(cos(z)+cos(3*z))/2*sin(2*z)+sin(4*z) при z=-1/3
- 2*(cos(z)+cos(3*z)) разделить на 2*sin(2*z)+sin(4*z) если z=-1 разделить на 3
-
Похожие выражения
- 2*(cos(z)+cos(3*z))/2*sin(2*z)-sin(4*z) если z=-1/3
- 2*(cos(z)-cos(3*z))/2*sin(2*z)+sin(4*z) если z=-1/3
- 2*(cos(z)+cos(3*z))/2*sin(2*z)+sin(4*z) если z=+1/3
2*(cos(z)+cos(3*z))/2*sin(2*z)+sin(4*z) если z=-1/3
Решение
Вы ввели
[src]
2*(cos(z) + cos(3*z))*sin(2*z)
------------------------------ + sin(4*z)
2
$$\frac{2 \left(\cos{\left(z \right)} + \cos{\left(3 z \right)}\right) \sin{\left(2 z \right)}}{2} + \sin{\left(4 z \right)}$$
2*(cos(z) + cos(3*z))*sin(2*z)/2 + sin(4*z)
Общее упрощение
[src]
sin(3*z) sin(5*z) -------- + -------- + sin(4*z) 2 2
$$\frac{\sin{\left(3 z \right)}}{2} + \sin{\left(4 z \right)} + \frac{\sin{\left(5 z \right)}}{2}$$
sin(3*z)/2 + sin(5*z)/2 + sin(4*z)
Подстановка условия
[src]
2*(cos(z) + cos(3*z))*sin(2*z)/2 + sin(4*z) при z = -1/3
подставляем
2*(cos(z) + cos(3*z))*sin(2*z)
------------------------------ + sin(4*z)
2
$$\frac{2 \left(\cos{\left(z \right)} + \cos{\left(3 z \right)}\right) \sin{\left(2 z \right)}}{2} + \sin{\left(4 z \right)}$$
sin(3*z) sin(5*z) -------- + -------- + sin(4*z) 2 2
$$\frac{\sin{\left(3 z \right)}}{2} + \sin{\left(4 z \right)} + \frac{\sin{\left(5 z \right)}}{2}$$
переменные
z = -1/3
$$z = - \frac{1}{3}$$
sin(3*(-1/3)) sin(5*(-1/3))
------------- + ------------- + sin(4*(-1/3))
2 2
$$\frac{\sin{\left(3 (-1/3) \right)}}{2} + \sin{\left(4 (-1/3) \right)} + \frac{\sin{\left(5 (-1/3) \right)}}{2}$$
sin(1) sin(5/3)
-sin(4/3) - ------ - --------
2 2
$$- \sin{\left(\frac{4}{3} \right)} - \frac{\sin{\left(\frac{5}{3} \right)}}{2} - \frac{\sin{\left(1 \right)}}{2}$$
-sin(4/3) - sin(1)/2 - sin(5/3)/2
Собрать выражение
[src]
sin(3*z) sin(5*z) -------- + -------- + sin(4*z) 2 2
$$\frac{\sin{\left(3 z \right)}}{2} + \sin{\left(4 z \right)} + \frac{\sin{\left(5 z \right)}}{2}$$
(cos(z) + cos(3*z))*sin(2*z) + sin(4*z)
$$\left(\cos{\left(z \right)} + \cos{\left(3 z \right)}\right) \sin{\left(2 z \right)} + \sin{\left(4 z \right)}$$
(cos(z) + cos(3*z))*sin(2*z) + sin(4*z)
Численный ответ
[src]
1.0*(cos(z) + cos(3*z))*sin(2*z) + sin(4*z)
1.0*(cos(z) + cos(3*z))*sin(2*z) + sin(4*z)
Рациональный знаменатель
[src]
cos(z)*sin(2*z) + cos(3*z)*sin(2*z) + sin(4*z)
$$\sin{\left(2 z \right)} \cos{\left(z \right)} + \sin{\left(2 z \right)} \cos{\left(3 z \right)} + \sin{\left(4 z \right)}$$
2*sin(4*z) + 2*(cos(z) + cos(3*z))*sin(2*z)
-------------------------------------------
2
$$\frac{2 \left(\cos{\left(z \right)} + \cos{\left(3 z \right)}\right) \sin{\left(2 z \right)} + 2 \sin{\left(4 z \right)}}{2}$$
(2*sin(4*z) + 2*(cos(z) + cos(3*z))*sin(2*z))/2
Комбинаторика
[src]
cos(z)*sin(2*z) + cos(3*z)*sin(2*z) + sin(4*z)
$$\sin{\left(2 z \right)} \cos{\left(z \right)} + \sin{\left(2 z \right)} \cos{\left(3 z \right)} + \sin{\left(4 z \right)}$$
cos(z)*sin(2*z) + cos(3*z)*sin(2*z) + sin(4*z)
Объединение рациональных выражений
[src]
(cos(z) + cos(3*z))*sin(2*z) + sin(4*z)
$$\left(\cos{\left(z \right)} + \cos{\left(3 z \right)}\right) \sin{\left(2 z \right)} + \sin{\left(4 z \right)}$$
(cos(z) + cos(3*z))*sin(2*z) + sin(4*z)
Степени
[src]
(cos(z) + cos(3*z))*sin(2*z) + sin(4*z)
$$\left(\cos{\left(z \right)} + \cos{\left(3 z \right)}\right) \sin{\left(2 z \right)} + \sin{\left(4 z \right)}$$
/ I*z -I*z -3*I*z 3*I*z\
/ -2*I*z 2*I*z\ |e e e e |
/ -4*I*z 4*I*z\ I*\- e + e /*|---- + ----- + ------- + ------|
I*\- e + e / \ 2 2 2 2 /
- ---------------------- - --------------------------------------------------------
2 2
$$- \frac{i \left(e^{2 i z} - e^{- 2 i z}\right) \left(\frac{e^{3 i z}}{2} + \frac{e^{i z}}{2} + \frac{e^{- i z}}{2} + \frac{e^{- 3 i z}}{2}\right)}{2} - \frac{i \left(e^{4 i z} - e^{- 4 i z}\right)}{2}$$
-i*(-exp(-4*i*z) + exp(4*i*z))/2 - i*(-exp(-2*i*z) + exp(2*i*z))*(exp(i*z)/2 + exp(-i*z)/2 + exp(-3*i*z)/2 + exp(3*i*z)/2)/2
Общий знаменатель
[src]
cos(z)*sin(2*z) + cos(3*z)*sin(2*z) + sin(4*z)
$$\sin{\left(2 z \right)} \cos{\left(z \right)} + \sin{\left(2 z \right)} \cos{\left(3 z \right)} + \sin{\left(4 z \right)}$$
cos(z)*sin(2*z) + cos(3*z)*sin(2*z) + sin(4*z)
Тригонометрическая часть
[src]
sin(3*z) sin(5*z) -------- + -------- + sin(4*z) 2 2
$$\frac{\sin{\left(3 z \right)}}{2} + \sin{\left(4 z \right)} + \frac{\sin{\left(5 z \right)}}{2}$$
2*cos(z)*cos(2*z)*sin(2*z) + sin(4*z)
$$2 \sin{\left(2 z \right)} \cos{\left(z \right)} \cos{\left(2 z \right)} + \sin{\left(4 z \right)}$$
1 1 1 -------- + ---------- + ---------- csc(4*z) 2*csc(3*z) 2*csc(5*z)
$$\frac{1}{2 \csc{\left(5 z \right)}} + \frac{1}{\csc{\left(4 z \right)}} + \frac{1}{2 \csc{\left(3 z \right)}}$$
(cos(z) + cos(3*z))*sin(2*z) + sin(4*z)
$$\left(\cos{\left(z \right)} + \cos{\left(3 z \right)}\right) \sin{\left(2 z \right)} + \sin{\left(4 z \right)}$$
1 1
------ + --------
1 sec(z) sec(3*z)
-------- + -----------------
csc(4*z) csc(2*z)
$$\frac{\frac{1}{\sec{\left(3 z \right)}} + \frac{1}{\sec{\left(z \right)}}}{\csc{\left(2 z \right)}} + \frac{1}{\csc{\left(4 z \right)}}$$
/ pi\ / pi\
(cos(z) + cos(3*z))*cos|2*z - --| + cos|4*z - --|
\ 2 / \ 2 /
$$\left(\cos{\left(z \right)} + \cos{\left(3 z \right)}\right) \cos{\left(2 z - \frac{\pi}{2} \right)} + \cos{\left(4 z - \frac{\pi}{2} \right)}$$
/ / pi\ /pi \\ |sin|z + --| + sin|-- + 3*z||*sin(2*z) + sin(4*z) \ \ 2 / \2 //
$$\left(\sin{\left(z + \frac{\pi}{2} \right)} + \sin{\left(3 z + \frac{\pi}{2} \right)}\right) \sin{\left(2 z \right)} + \sin{\left(4 z \right)}$$
/ pi\ / pi\
cos|3*z - --| cos|5*z - --|
\ 2 / \ 2 / / pi\
------------- + ------------- + cos|4*z - --|
2 2 \ 2 /
$$\frac{\cos{\left(3 z - \frac{\pi}{2} \right)}}{2} + \cos{\left(4 z - \frac{\pi}{2} \right)} + \frac{\cos{\left(5 z - \frac{\pi}{2} \right)}}{2}$$
1 1 1 ------------- + --------------- + --------------- / pi\ / pi\ / pi\ sec|4*z - --| 2*sec|3*z - --| 2*sec|5*z - --| \ 2 / \ 2 / \ 2 /
$$\frac{1}{2 \sec{\left(5 z - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(4 z - \frac{\pi}{2} \right)}} + \frac{1}{2 \sec{\left(3 z - \frac{\pi}{2} \right)}}$$
1 1
------ + --------
1 sec(z) sec(3*z)
------------- + -----------------
/ pi\ / pi\
sec|4*z - --| sec|2*z - --|
\ 2 / \ 2 /
$$\frac{\frac{1}{\sec{\left(3 z \right)}} + \frac{1}{\sec{\left(z \right)}}}{\sec{\left(2 z - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(4 z - \frac{\pi}{2} \right)}}$$
1 1
------ + --------
1 sec(z) sec(3*z)
------------- + -----------------
/pi \ /pi \
sec|-- - 4*z| sec|-- - 2*z|
\2 / \2 /
$$\frac{\frac{1}{\sec{\left(3 z \right)}} + \frac{1}{\sec{\left(z \right)}}}{\sec{\left(- 2 z + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(- 4 z + \frac{\pi}{2} \right)}}$$
1 1
----------- + -------------
/pi \ /pi \
csc|-- - z| csc|-- - 3*z|
1 \2 / \2 /
-------- + ---------------------------
csc(4*z) csc(2*z)
$$\frac{\frac{1}{\csc{\left(- z + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 3 z + \frac{\pi}{2} \right)}}}{\csc{\left(2 z \right)}} + \frac{1}{\csc{\left(4 z \right)}}$$
1 1
----------- + -------------
/pi \ /pi \
csc|-- - z| csc|-- - 3*z|
1 \2 / \2 /
------------- + ---------------------------
csc(pi - 4*z) csc(pi - 2*z)
$$\frac{\frac{1}{\csc{\left(- z + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 3 z + \frac{\pi}{2} \right)}}}{\csc{\left(- 2 z + \pi \right)}} + \frac{1}{\csc{\left(- 4 z + \pi \right)}}$$
/ 2 2 \ 2*(cos(z) + cos(3*z))*cos(z)*sin(z) + 4*\cos (z) - sin (z)/*cos(z)*sin(z)
$$4 \left(- \sin^{2}{\left(z \right)} + \cos^{2}{\left(z \right)}\right) \sin{\left(z \right)} \cos{\left(z \right)} + 2 \left(\cos{\left(z \right)} + \cos{\left(3 z \right)}\right) \sin{\left(z \right)} \cos{\left(z \right)}$$
/3*z\ /5*z\
tan|---| tan|---|
\ 2 / \ 2 / 2*tan(2*z)
------------- + ------------- + -------------
2/3*z\ 2/5*z\ 2
1 + tan |---| 1 + tan |---| 1 + tan (2*z)
\ 2 / \ 2 /
$$\frac{\tan{\left(\frac{5 z}{2} \right)}}{\tan^{2}{\left(\frac{5 z}{2} \right)} + 1} + \frac{2 \tan{\left(2 z \right)}}{\tan^{2}{\left(2 z \right)} + 1} + \frac{\tan{\left(\frac{3 z}{2} \right)}}{\tan^{2}{\left(\frac{3 z}{2} \right)} + 1}$$
/ 2/z\ 2/3*z\\
|1 - tan |-| 1 - tan |---||
| \2/ \ 2 /|
2*|----------- + -------------|*tan(z)
| 2/z\ 2/3*z\|
|1 + tan |-| 1 + tan |---||
2*tan(2*z) \ \2/ \ 2 //
------------- + --------------------------------------
2 2
1 + tan (2*z) 1 + tan (z)
$$\frac{2 \left(\frac{- \tan^{2}{\left(\frac{z}{2} \right)} + 1}{\tan^{2}{\left(\frac{z}{2} \right)} + 1} + \frac{- \tan^{2}{\left(\frac{3 z}{2} \right)} + 1}{\tan^{2}{\left(\frac{3 z}{2} \right)} + 1}\right) \tan{\left(z \right)}}{\tan^{2}{\left(z \right)} + 1} + \frac{2 \tan{\left(2 z \right)}}{\tan^{2}{\left(2 z \right)} + 1}$$
/ 0 for 3*z mod pi = 0 / 0 for 5*z mod pi = 0
< <
\sin(3*z) otherwise \sin(5*z) otherwise // 0 for 4*z mod pi = 0\
----------------------------- + ----------------------------- + |< |
2 2 \\sin(4*z) otherwise /
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 z \bmod \pi = 0 \\\sin{\left(3 z \right)} & \text{otherwise} \end{cases}}{2}\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\sin{\left(4 z \right)} & \text{otherwise} \end{cases}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: 5 z \bmod \pi = 0 \\\sin{\left(5 z \right)} & \text{otherwise} \end{cases}}{2}\right)$$
/ /z pi\ /pi 3*z\ \
| 2*tan|- + --| 2*tan|-- + ---| |
| \2 4 / \4 2 / |
2*|---------------- + ------------------|*cot(z)
| 2/z pi\ 2/pi 3*z\|
|1 + tan |- + --| 1 + tan |-- + ---||
2*cot(2*z) \ \2 4 / \4 2 //
------------- + ------------------------------------------------
2 2
1 + cot (2*z) 1 + cot (z)
$$\frac{2 \cdot \left(\frac{2 \tan{\left(\frac{3 z}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{3 z}{2} + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(\frac{z}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{z}{2} + \frac{\pi}{4} \right)} + 1}\right) \cot{\left(z \right)}}{\cot^{2}{\left(z \right)} + 1} + \frac{2 \cot{\left(2 z \right)}}{\cot^{2}{\left(2 z \right)} + 1}$$
/ /z pi\ /pi 3*z\ \
| 2*tan|- + --| 2*tan|-- + ---| |
| \2 4 / \4 2 / |
2*|---------------- + ------------------|*tan(z)
| 2/z pi\ 2/pi 3*z\|
|1 + tan |- + --| 1 + tan |-- + ---||
2*tan(2*z) \ \2 4 / \4 2 //
------------- + ------------------------------------------------
2 2
1 + tan (2*z) 1 + tan (z)
$$\frac{2 \cdot \left(\frac{2 \tan{\left(\frac{3 z}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{3 z}{2} + \frac{\pi}{4} \right)} + 1} + \frac{2 \tan{\left(\frac{z}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{z}{2} + \frac{\pi}{4} \right)} + 1}\right) \tan{\left(z \right)}}{\tan^{2}{\left(z \right)} + 1} + \frac{2 \tan{\left(2 z \right)}}{\tan^{2}{\left(2 z \right)} + 1}$$
/ 1 1 \
|1 - ------- 1 - ---------|
| 2/z\ 2/3*z\|
| cot |-| cot |---||
| \2/ \ 2 /|
2*|----------- + -------------|
| 1 1 |
|1 + ------- 1 + ---------|
| 2/z\ 2/3*z\|
| cot |-| cot |---||
2 \ \2/ \ 2 //
------------------------ + -------------------------------
/ 1 \ / 1 \
|1 + ---------|*cot(2*z) |1 + -------|*cot(z)
| 2 | | 2 |
\ cot (2*z)/ \ cot (z)/
$$\frac{2 \left(\frac{1 - \frac{1}{\cot^{2}{\left(\frac{z}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{z}{2} \right)}}} + \frac{1 - \frac{1}{\cot^{2}{\left(\frac{3 z}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{3 z}{2} \right)}}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(z \right)}}\right) \cot{\left(z \right)}} + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(2 z \right)}}\right) \cot{\left(2 z \right)}}$$
/ 2/z\ 2/3*z\\
|-1 + cot |-| -1 + cot |---||
/ 2/ pi\\ | \2/ \ 2 /|
|-1 + tan |z + --||*|------------ + --------------|
2/ pi\ \ \ 4 // | 2/z\ 2/3*z\ |
-1 + tan |2*z + --| |1 + cot |-| 1 + cot |---| |
\ 4 / \ \2/ \ 2 / /
------------------- + ---------------------------------------------------
2/ pi\ 2/ pi\
1 + tan |2*z + --| 1 + tan |z + --|
\ 4 / \ 4 /
$$\frac{\left(\frac{\cot^{2}{\left(\frac{z}{2} \right)} - 1}{\cot^{2}{\left(\frac{z}{2} \right)} + 1} + \frac{\cot^{2}{\left(\frac{3 z}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 z}{2} \right)} + 1}\right) \left(\tan^{2}{\left(z + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(z + \frac{\pi}{4} \right)} + 1} + \frac{\tan^{2}{\left(2 z + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(2 z + \frac{\pi}{4} \right)} + 1}$$
/ 2/z\ 2/3*z\\
|1 - tan |-| 1 - tan |---||
/ 2/ pi\\ | \2/ \ 2 /|
|1 - cot |z + --||*|----------- + -------------|
2/ pi\ \ \ 4 // | 2/z\ 2/3*z\|
1 - cot |2*z + --| |1 + tan |-| 1 + tan |---||
\ 4 / \ \2/ \ 2 //
------------------ + ------------------------------------------------
2/ pi\ 2/ pi\
1 + cot |2*z + --| 1 + cot |z + --|
\ 4 / \ 4 /
$$\frac{\left(- \cot^{2}{\left(z + \frac{\pi}{4} \right)} + 1\right) \left(\frac{- \tan^{2}{\left(\frac{z}{2} \right)} + 1}{\tan^{2}{\left(\frac{z}{2} \right)} + 1} + \frac{- \tan^{2}{\left(\frac{3 z}{2} \right)} + 1}{\tan^{2}{\left(\frac{3 z}{2} \right)} + 1}\right)}{\cot^{2}{\left(z + \frac{\pi}{4} \right)} + 1} + \frac{- \cot^{2}{\left(2 z + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(2 z + \frac{\pi}{4} \right)} + 1}$$
2 2 / 2*(-1 - cos(2*z) + 2*cos(z)) 2*(-1 - cos(6*z) + 2*cos(3*z)) \
4*cos (z)*sin (z)*|------------------------------ + --------------------------------|
| 2 2|
\1 - cos(2*z) + 2*(1 - cos(z)) 1 - cos(6*z) + 2*(1 - cos(3*z)) /
------------------------------------------------------------------------------------- + sin(4*z)
sin(2*z)
$$\frac{4 \cdot \left(\frac{2 \cdot \left(2 \cos{\left(3 z \right)} - \cos{\left(6 z \right)} - 1\right)}{2 \left(- \cos{\left(3 z \right)} + 1\right)^{2} - \cos{\left(6 z \right)} + 1} + \frac{2 \cdot \left(2 \cos{\left(z \right)} - \cos{\left(2 z \right)} - 1\right)}{2 \left(- \cos{\left(z \right)} + 1\right)^{2} - \cos{\left(2 z \right)} + 1}\right) \sin^{2}{\left(z \right)} \cos^{2}{\left(z \right)}}{\sin{\left(2 z \right)}} + \sin{\left(4 z \right)}$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\ ||< | + |< ||*|< | + |< | \\\cos(z) otherwise / \\cos(3*z) otherwise // \\sin(2*z) otherwise / \\sin(4*z) otherwise /
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\cos{\left(z \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\cos{\left(3 z \right)} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\sin{\left(2 z \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\sin{\left(4 z \right)} & \text{otherwise} \end{cases}\right)$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ ||| | || || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\ ||< / pi\ | + |< /pi \ ||*|< | + |< | |||sin|z + --| otherwise | ||sin|-- + 3*z| otherwise || \\sin(2*z) otherwise / \\sin(4*z) otherwise / \\\ \ 2 / / \\ \2 / //
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\sin{\left(z + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\sin{\left(3 z + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\sin{\left(2 z \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\sin{\left(4 z \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ || | || |
||< | + |< ||*|< / pi\ | + |< / pi\ |
\\\cos(z) otherwise / \\cos(3*z) otherwise // ||cos|2*z - --| otherwise | ||cos|4*z - --| otherwise |
\\ \ 2 / / \\ \ 2 / /
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\cos{\left(z \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\cos{\left(3 z \right)} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\cos{\left(2 z - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\cos{\left(4 z - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
/ 0 for 3*z mod pi = 0 / 0 for 5*z mod pi = 0
| |
| /3*z\ | /5*z\
| 2*cot|---| | 2*cot|---|
< \ 2 / < \ 2 /
|------------- otherwise |------------- otherwise
| 2/3*z\ | 2/5*z\ // 0 for 4*z mod pi = 0\
|1 + cot |---| |1 + cot |---| || |
\ \ 2 / \ \ 2 / || 2*cot(2*z) |
---------------------------------- + ---------------------------------- + |<------------- otherwise |
2 2 || 2 |
||1 + cot (2*z) |
\\ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: 3 z \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 z}{2} \right)}}{\cot^{2}{\left(\frac{3 z}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2}\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{2 \cot{\left(2 z \right)}}{\cot^{2}{\left(2 z \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: 5 z \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{5 z}{2} \right)}}{\cot^{2}{\left(\frac{5 z}{2} \right)} + 1} & \text{otherwise} \end{cases}}{2}\right)$$
// / 3*pi\ \ // / 3*pi\ \
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ || 1 for |2*z + ----| mod 2*pi = 0| || 1 for |4*z + ----| mod 2*pi = 0|
||< | + |< ||*|< \ 2 / | + |< \ 2 / |
\\\cos(z) otherwise / \\cos(3*z) otherwise // || | || |
\\sin(2*z) otherwise / \\sin(4*z) otherwise /
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\cos{\left(z \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\cos{\left(3 z \right)} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 1 & \text{for}\: \left(2 z + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 z \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(4 z + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(4 z \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ || | || |
||| | || || || 1 | || 1 |
||< 1 | + |< 1 ||*|<------------- otherwise | + |<------------- otherwise |
|||------ otherwise | ||-------- otherwise || || / pi\ | || / pi\ |
\\\sec(z) / \\sec(3*z) // ||sec|2*z - --| | ||sec|4*z - --| |
\\ \ 2 / / \\ \ 2 / /
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(z \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(3 z \right)}} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\frac{1}{\sec{\left(2 z - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{1}{\sec{\left(4 z - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ ||| | || || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\ ||| 1 | || 1 || || | || | ||<----------- otherwise | + |<------------- otherwise ||*|< 1 | + |< 1 | ||| /pi \ | || /pi \ || ||-------- otherwise | ||-------- otherwise | |||csc|-- - z| | ||csc|-- - 3*z| || \\csc(2*z) / \\csc(4*z) / \\\ \2 / / \\ \2 / //
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- z + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 3 z + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\frac{1}{\csc{\left(2 z \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{1}{\csc{\left(4 z \right)}} & \text{otherwise} \end{cases}\right)$$
/ 4/z\ 4/3*z\\
| 4*sin |-| 4*sin |---||
| \2/ \ 2 /|
|1 - --------- 1 - -----------|
| 2 2 |
2 | sin (z) sin (3*z) |
4*sin (z)*|------------- + ---------------|
| 4/z\ 4/3*z\|
| 4*sin |-| 4*sin |---||
| \2/ \ 2 /|
|1 + --------- 1 + -----------|
2 | 2 2 |
4*sin (2*z) \ sin (z) sin (3*z) /
-------------------------- + -------------------------------------------
/ 4 \ / 4 \
| 4*sin (2*z)| | 4*sin (z)|
|1 + -----------|*sin(4*z) |1 + ---------|*sin(2*z)
| 2 | | 2 |
\ sin (4*z) / \ sin (2*z)/
$$\frac{4 \left(\frac{- \frac{4 \sin^{4}{\left(\frac{z}{2} \right)}}{\sin^{2}{\left(z \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{z}{2} \right)}}{\sin^{2}{\left(z \right)}} + 1} + \frac{- \frac{4 \sin^{4}{\left(\frac{3 z}{2} \right)}}{\sin^{2}{\left(3 z \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{3 z}{2} \right)}}{\sin^{2}{\left(3 z \right)}} + 1}\right) \sin^{2}{\left(z \right)}}{\left(\frac{4 \sin^{4}{\left(z \right)}}{\sin^{2}{\left(2 z \right)}} + 1\right) \sin{\left(2 z \right)}} + \frac{4 \sin^{2}{\left(2 z \right)}}{\left(\frac{4 \sin^{4}{\left(2 z \right)}}{\sin^{2}{\left(4 z \right)}} + 1\right) \sin{\left(4 z \right)}}$$
/// / pi\ \ // /pi \ \\ ||| 0 for |z + --| mod pi = 0| || 0 for |-- + 3*z| mod pi = 0|| ||| \ 2 / | || \2 / || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\ ||< | + |< ||*|< | + |< | ||| /z pi\ | || /pi 3*z\ || \\sin(2*z) otherwise / \\sin(4*z) otherwise / |||(1 + sin(z))*cot|- + --| otherwise | ||(1 + sin(3*z))*cot|-- + ---| otherwise || \\\ \2 4 / / \\ \4 2 / //
$$\left(\left(\left(\begin{cases} 0 & \text{for}\: \left(z + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(z \right)} + 1\right) \cot{\left(\frac{z}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(3 z + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(3 z \right)} + 1\right) \cot{\left(\frac{3 z}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\sin{\left(2 z \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\sin{\left(4 z \right)} & \text{otherwise} \end{cases}\right)$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ ||| | || || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\ ||| 2/z\ | || 2/3*z\ || || | || | |||-1 + cot |-| | ||-1 + cot |---| || || 2*cot(z) | || 2*cot(2*z) | ||< \2/ | + |< \ 2 / ||*|<----------- otherwise | + |<------------- otherwise | |||------------ otherwise | ||-------------- otherwise || || 2 | || 2 | ||| 2/z\ | || 2/3*z\ || ||1 + cot (z) | ||1 + cot (2*z) | |||1 + cot |-| | ||1 + cot |---| || \\ / \\ / \\\ \2/ / \\ \ 2 / //
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{z}{2} \right)} - 1}{\cot^{2}{\left(\frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 z}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 z}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\frac{2 \cot{\left(z \right)}}{\cot^{2}{\left(z \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{2 \cot{\left(2 z \right)}}{\cot^{2}{\left(2 z \right)} + 1} & \text{otherwise} \end{cases}\right)$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ ||| | || || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\ ||| 2/z\ | || 2/3*z\ || || | || | |||1 - tan |-| | ||1 - tan |---| || || 2*tan(z) | || 2*tan(2*z) | ||< \2/ | + |< \ 2 / ||*|<----------- otherwise | + |<------------- otherwise | |||----------- otherwise | ||------------- otherwise || || 2 | || 2 | ||| 2/z\ | || 2/3*z\ || ||1 + tan (z) | ||1 + tan (2*z) | |||1 + tan |-| | ||1 + tan |---| || \\ / \\ / \\\ \2/ / \\ \ 2 / //
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{z}{2} \right)} + 1}{\tan^{2}{\left(\frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{3 z}{2} \right)} + 1}{\tan^{2}{\left(\frac{3 z}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\frac{2 \tan{\left(z \right)}}{\tan^{2}{\left(z \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{2 \tan{\left(2 z \right)}}{\tan^{2}{\left(2 z \right)} + 1} & \text{otherwise} \end{cases}\right)$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ ||| | || || ||| 2 | || -2 - 2*cos(6*z) + 4*cos(3*z) || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\ ||< -4 + 4*sin (z) + 4*cos(z) | + |<-------------------------------- otherwise ||*|< | + |< | |||--------------------------- otherwise | || 2 || \\sin(2*z) otherwise / \\sin(4*z) otherwise / ||| 2 2 | ||1 - cos(6*z) + 2*(1 - cos(3*z)) || \\\2*(1 - cos(z)) + 2*sin (z) / \\ //
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(z \right)} + 4 \cos{\left(z \right)} - 4}{2 \left(- \cos{\left(z \right)} + 1\right)^{2} + 2 \sin^{2}{\left(z \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{4 \cos{\left(3 z \right)} - 2 \cos{\left(6 z \right)} - 2}{2 \left(- \cos{\left(3 z \right)} + 1\right)^{2} - \cos{\left(6 z \right)} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\sin{\left(2 z \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\sin{\left(4 z \right)} & \text{otherwise} \end{cases}\right)$$
/ 2/z pi\ 2/ pi 3*z\\
| cos |- - --| cos |- -- + ---||
| \2 2 / \ 2 2 /|
|1 - ------------ 1 - ----------------|
| 2/z\ 2/3*z\ |
| cos |-| cos |---| |
| \2/ \ 2 / | / pi\
2*|---------------- + --------------------|*cos|z - --|
| 2/z pi\ 2/ pi 3*z\| \ 2 /
| cos |- - --| cos |- -- + ---||
| \2 2 / \ 2 2 /|
|1 + ------------ 1 + ----------------|
/ pi\ | 2/z\ 2/3*z\ |
2*cos|2*z - --| | cos |-| cos |---| |
\ 2 / \ \2/ \ 2 / /
----------------------------- + -------------------------------------------------------
/ 2/ pi\\ / 2/ pi\\
| cos |2*z - --|| | cos |z - --||
| \ 2 /| | \ 2 /|
|1 + --------------|*cos(2*z) |1 + ------------|*cos(z)
| 2 | | 2 |
\ cos (2*z) / \ cos (z) /
$$\frac{2 \left(\frac{1 - \frac{\cos^{2}{\left(\frac{z}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{z}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{z}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{z}{2} \right)}}} + \frac{1 - \frac{\cos^{2}{\left(\frac{3 z}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 z}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{3 z}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 z}{2} \right)}}}\right) \cos{\left(z - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(z - \frac{\pi}{2} \right)}}{\cos^{2}{\left(z \right)}}\right) \cos{\left(z \right)}} + \frac{2 \cos{\left(2 z - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(2 z - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 z \right)}}\right) \cos{\left(2 z \right)}}$$
/ 2/z\ 2/3*z\ \
| sec |-| sec |---| |
| \2/ \ 2 / |
|1 - ------------ 1 - ----------------|
| 2/z pi\ 2/ pi 3*z\|
| sec |- - --| sec |- -- + ---||
| \2 2 / \ 2 2 /|
2*|---------------- + --------------------|*sec(z)
| 2/z\ 2/3*z\ |
| sec |-| sec |---| |
| \2/ \ 2 / |
|1 + ------------ 1 + ----------------|
| 2/z pi\ 2/ pi 3*z\|
| sec |- - --| sec |- -- + ---||
2*sec(2*z) \ \2 2 / \ 2 2 //
---------------------------------- + --------------------------------------------------
/ 2 \ / 2 \
| sec (2*z) | / pi\ | sec (z) | / pi\
|1 + --------------|*sec|2*z - --| |1 + ------------|*sec|z - --|
| 2/ pi\| \ 2 / | 2/ pi\| \ 2 /
| sec |2*z - --|| | sec |z - --||
\ \ 2 // \ \ 2 //
$$\frac{2 \left(\frac{- \frac{\sec^{2}{\left(\frac{z}{2} \right)}}{\sec^{2}{\left(\frac{z}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{z}{2} \right)}}{\sec^{2}{\left(\frac{z}{2} - \frac{\pi}{2} \right)}} + 1} + \frac{- \frac{\sec^{2}{\left(\frac{3 z}{2} \right)}}{\sec^{2}{\left(\frac{3 z}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{3 z}{2} \right)}}{\sec^{2}{\left(\frac{3 z}{2} - \frac{\pi}{2} \right)}} + 1}\right) \sec{\left(z \right)}}{\left(\frac{\sec^{2}{\left(z \right)}}{\sec^{2}{\left(z - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(z - \frac{\pi}{2} \right)}} + \frac{2 \sec{\left(2 z \right)}}{\left(\frac{\sec^{2}{\left(2 z \right)}}{\sec^{2}{\left(2 z - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(2 z - \frac{\pi}{2} \right)}}$$
/ 2/pi z\ 2/pi 3*z\\
| csc |-- - -| csc |-- - ---||
| \2 2/ \2 2 /|
|1 - ------------ 1 - --------------|
| 2/z\ 2/3*z\ |
| csc |-| csc |---| |
| \2/ \ 2 / | /pi \
2*|---------------- + ------------------|*csc|-- - z|
| 2/pi z\ 2/pi 3*z\| \2 /
| csc |-- - -| csc |-- - ---||
| \2 2/ \2 2 /|
|1 + ------------ 1 + --------------|
/pi \ | 2/z\ 2/3*z\ |
2*csc|-- - 2*z| | csc |-| csc |---| |
\2 / \ \2/ \ 2 / /
----------------------------- + -----------------------------------------------------
/ 2/pi \\ / 2/pi \\
| csc |-- - 2*z|| | csc |-- - z||
| \2 /| | \2 /|
|1 + --------------|*csc(2*z) |1 + ------------|*csc(z)
| 2 | | 2 |
\ csc (2*z) / \ csc (z) /
$$\frac{2 \left(\frac{1 - \frac{\csc^{2}{\left(- \frac{z}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{z}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{z}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{z}{2} \right)}}} + \frac{1 - \frac{\csc^{2}{\left(- \frac{3 z}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 z}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{3 z}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 z}{2} \right)}}}\right) \csc{\left(- z + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- z + \frac{\pi}{2} \right)}}{\csc^{2}{\left(z \right)}}\right) \csc{\left(z \right)}} + \frac{2 \csc{\left(- 2 z + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- 2 z + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 z \right)}}\right) \csc{\left(2 z \right)}}$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\
||| | || ||
||| 1 | || 1 || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\
|||-1 + ------- | ||-1 + --------- || || | || |
||| 2/z\ | || 2/3*z\ || || 2 | || 2 |
||| tan |-| | || tan |---| || ||-------------------- otherwise | ||------------------------ otherwise |
||< \2/ | + |< \ 2 / ||*|
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{z}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{z}{2} \right)}}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{3 z}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{3 z}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(z \right)}}\right) \tan{\left(z \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(2 z \right)}}\right) \tan{\left(2 z \right)}} & \text{otherwise} \end{cases}\right)$$
/// / pi\ \ // /pi \ \\ ||| 0 for |z + --| mod pi = 0| || 0 for |-- + 3*z| mod pi = 0|| ||| \ 2 / | || \2 / || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\ ||| | || || || | || | ||| /z pi\ | || /pi 3*z\ || || 2*cot(z) | || 2*cot(2*z) | ||< 2*cot|- + --| | + |< 2*cot|-- + ---| ||*|<----------- otherwise | + |<------------- otherwise | ||| \2 4 / | || \4 2 / || || 2 | || 2 | |||---------------- otherwise | ||------------------ otherwise || ||1 + cot (z) | ||1 + cot (2*z) | ||| 2/z pi\ | || 2/pi 3*z\ || \\ / \\ / |||1 + cot |- + --| | ||1 + cot |-- + ---| || \\\ \2 4 / / \\ \4 2 / //
$$\left(\left(\left(\begin{cases} 0 & \text{for}\: \left(z + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{z}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{z}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(3 z + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 z}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{3 z}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\frac{2 \cot{\left(z \right)}}{\cot^{2}{\left(z \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{2 \cot{\left(2 z \right)}}{\cot^{2}{\left(2 z \right)} + 1} & \text{otherwise} \end{cases}\right)$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\
||| | || || || | || |
||
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\cos{\left(z \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\cos{\left(3 z \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\sin{\left(2 z \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\sin{\left(4 z \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \ // / 3*pi\ \
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ || 1 for |2*z + ----| mod 2*pi = 0| || 1 for |4*z + ----| mod 2*pi = 0|
||| | || || || \ 2 / | || \ 2 / |
||| 2/z\ | || 2/3*z\ || || | || |
|||-1 + cot |-| | ||-1 + cot |---| || || 2/ pi\ | || 2/ pi\ |
||< \2/ | + |< \ 2 / ||*|<-1 + tan |z + --| | + |<-1 + tan |2*z + --| |
|||------------ otherwise | ||-------------- otherwise || || \ 4 / | || \ 4 / |
||| 2/z\ | || 2/3*z\ || ||----------------- otherwise | ||------------------- otherwise |
|||1 + cot |-| | ||1 + cot |---| || || 2/ pi\ | || 2/ pi\ |
\\\ \2/ / \\ \ 2 / // || 1 + tan |z + --| | || 1 + tan |2*z + --| |
\\ \ 4 / / \\ \ 4 / /
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{z}{2} \right)} - 1}{\cot^{2}{\left(\frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 z}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 z}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 1 & \text{for}\: \left(2 z + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(z + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(z + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 1 & \text{for}\: \left(4 z + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(2 z + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(2 z + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\
||| | || ||
||| 2 | || 2 ||
||| sin (z) | || sin (3*z) || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\
|||-1 + --------- | ||-1 + ----------- || || | || |
||| 4/z\ | || 4/3*z\ || || sin(2*z) | || sin(4*z) |
||| 4*sin |-| | || 4*sin |---| || ||----------------------- otherwise | ||--------------------------- otherwise |
||< \2/ | + |< \ 2 / ||*|
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(z \right)}}{4 \sin^{4}{\left(\frac{z}{2} \right)}}}{1 + \frac{\sin^{2}{\left(z \right)}}{4 \sin^{4}{\left(\frac{z}{2} \right)}}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(3 z \right)}}{4 \sin^{4}{\left(\frac{3 z}{2} \right)}}}{1 + \frac{\sin^{2}{\left(3 z \right)}}{4 \sin^{4}{\left(\frac{3 z}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\frac{\sin{\left(2 z \right)}}{\left(1 + \frac{\sin^{2}{\left(2 z \right)}}{4 \sin^{4}{\left(z \right)}}\right) \sin^{2}{\left(z \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{\sin{\left(4 z \right)}}{\left(1 + \frac{\sin^{2}{\left(4 z \right)}}{4 \sin^{4}{\left(2 z \right)}}\right) \sin^{2}{\left(2 z \right)}} & \text{otherwise} \end{cases}\right)$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ ||| | || || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\ |||/ 1 for z mod 2*pi = 0 | ||/ 1 for 3*z mod 2*pi = 0 || || | || | |||| | ||| || ||/ 0 for 2*z mod pi = 0 | ||/ 0 for 4*z mod pi = 0 | |||| 2/z\ | ||| 2/3*z\ || ||| | ||| | ||<|-1 + cot |-| | + |<|-1 + cot |---| ||*|<| 2*cot(z) | + |<| 2*cot(2*z) | |||< \2/ otherwise | ||< \ 2 / otherwise || ||<----------- otherwise otherwise | ||<------------- otherwise otherwise | ||||------------ otherwise | |||-------------- otherwise || ||| 2 | ||| 2 | |||| 2/z\ | ||| 2/3*z\ || |||1 + cot (z) | |||1 + cot (2*z) | ||||1 + cot |-| | |||1 + cot |---| || \\\ / \\\ / \\\\ \2/ / \\\ \ 2 / //
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{z}{2} \right)} - 1}{\cot^{2}{\left(\frac{z}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 z}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 z}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\frac{2 \cot{\left(z \right)}}{\cot^{2}{\left(z \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{2 \cot{\left(2 z \right)}}{\cot^{2}{\left(2 z \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ ||| | || || ||| 2/z\ | || 2/3*z\ || ||| cos |-| | || cos |---| || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\ ||| \2/ | || \ 2 / || || | || | |||-1 + ------------ | ||-1 + ---------------- || || 2*cos(z) | || 2*cos(2*z) | ||| 2/z pi\ | || 2/ pi 3*z\ || ||------------------------------ otherwise | ||---------------------------------- otherwise | ||| cos |- - --| | || cos |- -- + ---| || ||/ 2 \ | ||/ 2 \ | ||< \2 2 / | + |< \ 2 2 / ||*|<| cos (z) | / pi\ | + |<| cos (2*z) | / pi\ | |||----------------- otherwise | ||--------------------- otherwise || |||1 + ------------|*cos|z - --| | |||1 + --------------|*cos|2*z - --| | ||| 2/z\ | || 2/3*z\ || ||| 2/ pi\| \ 2 / | ||| 2/ pi\| \ 2 / | ||| cos |-| | || cos |---| || ||| cos |z - --|| | ||| cos |2*z - --|| | ||| \2/ | || \ 2 / || ||\ \ 2 // | ||\ \ 2 // | ||| 1 + ------------ | || 1 + ---------------- || \\ / \\ / ||| 2/z pi\ | || 2/ pi 3*z\ || ||| cos |- - --| | || cos |- -- + ---| || \\\ \2 2 / / \\ \ 2 2 / //
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{z}{2} \right)}}{\cos^{2}{\left(\frac{z}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{z}{2} \right)}}{\cos^{2}{\left(\frac{z}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{3 z}{2} \right)}}{\cos^{2}{\left(\frac{3 z}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{3 z}{2} \right)}}{\cos^{2}{\left(\frac{3 z}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\frac{2 \cos{\left(z \right)}}{\left(\frac{\cos^{2}{\left(z \right)}}{\cos^{2}{\left(z - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(z - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{2 \cos{\left(2 z \right)}}{\left(\frac{\cos^{2}{\left(2 z \right)}}{\cos^{2}{\left(2 z - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(2 z - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\
||| | || ||
||| 2/z pi\ | || 2/ pi 3*z\ || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\
||| sec |- - --| | || sec |- -- + ---| || || | || |
||| \2 2 / | || \ 2 2 / || || / pi\ | || / pi\ |
|||-1 + ------------ | ||-1 + ---------------- || || 2*sec|z - --| | || 2*sec|2*z - --| |
||| 2/z\ | || 2/3*z\ || || \ 2 / | || \ 2 / |
||| sec |-| | || sec |---| || ||------------------------- otherwise | ||----------------------------- otherwise |
||< \2/ | + |< \ 2 / ||*|
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{z}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{z}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{z}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{z}{2} \right)}}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{3 z}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 z}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{3 z}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 z}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\frac{2 \sec{\left(z - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(z - \frac{\pi}{2} \right)}}{\sec^{2}{\left(z \right)}}\right) \sec{\left(z \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{2 \sec{\left(2 z - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(2 z - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 z \right)}}\right) \sec{\left(2 z \right)}} & \text{otherwise} \end{cases}\right)$$
/// 1 for z mod 2*pi = 0\ // 1 for 3*z mod 2*pi = 0\\ ||| | || || ||| 2/z\ | || 2/3*z\ || ||| csc |-| | || csc |---| || // 0 for 2*z mod pi = 0\ // 0 for 4*z mod pi = 0\ ||| \2/ | || \ 2 / || || | || | |||-1 + ------------ | ||-1 + -------------- || || 2*csc(z) | || 2*csc(2*z) | ||| 2/pi z\ | || 2/pi 3*z\ || ||------------------------------ otherwise | ||---------------------------------- otherwise | ||| csc |-- - -| | || csc |-- - ---| || ||/ 2 \ | ||/ 2 \ | ||< \2 2/ | + |< \2 2 / ||*|<| csc (z) | /pi \ | + |<| csc (2*z) | /pi \ | |||----------------- otherwise | ||------------------- otherwise || |||1 + ------------|*csc|-- - z| | |||1 + --------------|*csc|-- - 2*z| | ||| 2/z\ | || 2/3*z\ || ||| 2/pi \| \2 / | ||| 2/pi \| \2 / | ||| csc |-| | || csc |---| || ||| csc |-- - z|| | ||| csc |-- - 2*z|| | ||| \2/ | || \ 2 / || ||\ \2 // | ||\ \2 // | ||| 1 + ------------ | || 1 + -------------- || \\ / \\ / ||| 2/pi z\ | || 2/pi 3*z\ || ||| csc |-- - -| | || csc |-- - ---| || \\\ \2 2/ / \\ \2 2 / //
$$\left(\left(\left(\begin{cases} 1 & \text{for}\: z \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{z}{2} \right)}}{\csc^{2}{\left(- \frac{z}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{z}{2} \right)}}{\csc^{2}{\left(- \frac{z}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: 3 z \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{3 z}{2} \right)}}{\csc^{2}{\left(- \frac{3 z}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{3 z}{2} \right)}}{\csc^{2}{\left(- \frac{3 z}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} 0 & \text{for}\: 2 z \bmod \pi = 0 \\\frac{2 \csc{\left(z \right)}}{\left(\frac{\csc^{2}{\left(z \right)}}{\csc^{2}{\left(- z + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- z + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\begin{cases} 0 & \text{for}\: 4 z \bmod \pi = 0 \\\frac{2 \csc{\left(2 z \right)}}{\left(\frac{\csc^{2}{\left(2 z \right)}}{\csc^{2}{\left(- 2 z + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 2 z + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
(Piecewise((1, Mod(z = 2*pi, 0)), ((-1 + csc(z/2)^2/csc(pi/2 - z/2)^2)/(1 + csc(z/2)^2/csc(pi/2 - z/2)^2), True)) + Piecewise((1, Mod(3*z = 2*pi, 0)), ((-1 + csc(3*z/2)^2/csc(pi/2 - 3*z/2)^2)/(1 + csc(3*z/2)^2/csc(pi/2 - 3*z/2)^2), True)))*Piecewise((0, Mod(2*z = pi, 0)), (2*csc(z)/((1 + csc(z)^2/csc(pi/2 - z)^2)*csc(pi/2 - z)), True)) + Piecewise((0, Mod(4*z = pi, 0)), (2*csc(2*z)/((1 + csc(2*z)^2/csc(pi/2 - 2*z)^2)*csc(pi/2 - 2*z)), True))